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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 13th 2011
   • (edited Jan 13th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI want to fix and harmonize some terminology and $n$Lab-entry titles and want to ask you all for your tastes and opinions. Now that I am more or less through with polishing <a href="http://ncatlab.org/nlab/show/infinity-Chern-Weil+theory+introduction">Part I, Introduction</a> and <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos">Part II, General abstract theory</a> for differential cohomology in cohesive $\infty$-toposes, I am about to turn to polishing _Part III: Models_ . For the time being this is supposed to consist of discussion of the three examples induced by the three $\infty$-cohesive pregeometries $CartSp_{Top}$, $CartSp_{smooth}$ and [[ThCartSp]]. (By the way, recently sombody indicated to me that it is a shame that not more than these three sites and their induced geometry enjoy that much coverage on the nLab. I fully agree, but here is a limit to the hours in a day. Hopefully some day we'll have more on other geometries, too.) I had started out calling the objects in the cohesive $\infty$-topos over $CartSp$ "$\infty$-Lie groupoids". This had always been met with understandable reservation, since these 1-truncated such gadgets can be considerbaly more general than what is strictly speaking called a _Lie groupoid_ . I had always thought and argued as follows, though: What matters is not the traditional notion of _Lie groupoid_ , what matters is _Lie theory_ . To make Lie theory come out fully nicely, standard Lie groupoids are too restrictive. But the cohesive $\infty$-topos over $CartSp$ seems to admit fully fledged generalized Lie theory So therefore it should be named after Lie. I still think that makes sense. On the other hand, in the same vein I had at one point titled the entry about the cohesive $\infty$-topos over $CartSp_{Top}$ [[topological infinity-groupoids]] (so far a stub!). That, too, led to some opposition. Notably Mike had complained about it. I understand all these complaints. But any terminology is always going to conflict with _some_ other terminology, so it's not a matter of true of false, but of taste. Still, I may have changed my mind now. My taste now is that I would like to have terminology follow the template "cohesive $\infty$-groupoid" by replacing "cohesive" with the respective _notion of cohesiveness_ . And the "cohesive structure" that characterizes Lie groupoids is not "Lie structure" but "smooth structure". So now I am inclined to rename that entry into "smooth $infty$-groupoid" . The cohesive structure encoded by $ThCartS$ I thought might well be called "synthetic differential structure" (If you see what I mean. But what do you think?). What about the case $CartSp_{top}$ (Cartesian spaces with continuous maps between them)? How about _continuous $\infty$-groupoids_ ? That's what I am curently tending towards: 1. "[[continuous infinity-groupoid]]" 1. "[[smooth infinity-groupoid]]" 1. "[[synthetic differential infinity-groupoid]]" Any opinions?

   I want to fix and harmonize some terminology and $n$Lab-entry titles and want to ask you all for your tastes and opinions.

   Now that I am more or less through with polishing Part I, Introduction and Part II, General abstract theory for differential cohomology in cohesive $\infty$-toposes, I am about to turn to polishing Part III: Models . For the time being this is supposed to consist of discussion of the three examples induced by the three $\infty$-cohesive pregeometries $CartSp_{Top}$, $CartSp_{smooth}$ and ThCartSp.

   (By the way, recently sombody indicated to me that it is a shame that not more than these three sites and their induced geometry enjoy that much coverage on the nLab. I fully agree, but here is a limit to the hours in a day. Hopefully some day we’ll have more on other geometries, too.)

   I had started out calling the objects in the cohesive $\infty$-topos over $CartSp$ “$\infty$-Lie groupoids”. This had always been met with understandable reservation, since these 1-truncated such gadgets can be considerbaly more general than what is strictly speaking called a Lie groupoid . I had always thought and argued as follows, though: What matters is not the traditional notion of Lie groupoid , what matters is Lie theory . To make Lie theory come out fully nicely, standard Lie groupoids are too restrictive. But the cohesive $\infty$-topos over $CartSp$ seems to admit fully fledged generalized Lie theory So therefore it should be named after Lie.

   I still think that makes sense. On the other hand, in the same vein I had at one point titled the entry about the cohesive $\infty$-topos over $CartSp_{Top}$ topological infinity-groupoids (so far a stub!). That, too, led to some opposition. Notably Mike had complained about it.

   I understand all these complaints. But any terminology is always going to conflict with some other terminology, so it’s not a matter of true of false, but of taste.

   Still, I may have changed my mind now. My taste now is that I would like to have terminology follow the template “cohesive $\infty$-groupoid” by replacing “cohesive” with the respective notion of cohesiveness . And the “cohesive structure” that characterizes Lie groupoids is not “Lie structure” but “smooth structure”. So now I am inclined to rename that entry into “smooth $infty$-groupoid” . The cohesive structure encoded by $ThCartS$ I thought might well be called “synthetic differential structure” (If you see what I mean. But what do you think?). What about the case $CartSp_{top}$ (Cartesian spaces with continuous maps between them)? How about continuous $\infty$-groupoids ?

   That’s what I am curently tending towards:

   1. “continuous infinity-groupoid”

   2. “smooth infinity-groupoid”

   3. “synthetic differential infinity-groupoid”

   Any opinions?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 13th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexAlso the term "discrete $\infty$-groupoid" (for a plain $\infty$-groupoid emphasing that any other cohesive structure that might be in the game is being disregarded) would fit in nicely. This is cohesive for the _discrete_ cohesive structure.

   Also the term “discrete $\infty$-groupoid” (for a plain $\infty$-groupoid emphasing that any other cohesive structure that might be in the game is being disregarded) would fit in nicely. This is cohesive for the discrete cohesive structure.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 13th 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItex>(By the way, recently sombody indicated to me that it is a shame that not more than these three sites and their induced geometry enjoy that much coverage on the nLab. I fully agree, but here is a limit to the hours in a day. Hopefully some day we'll have more on other geometries, too.) Does that mean they have more examples to add [here](http://ncatlab.org/nlab/show/cohesive+%28infinity,1%29-topos#Examples)? I saw your [call](http://mathoverflow.net/questions/42911/cohesive-toposes) for more examples had no response.

   (By the way, recently sombody indicated to me that it is a shame that not more than these three sites and their induced geometry enjoy that much coverage on the nLab. I fully agree, but here is a limit to the hours in a day. Hopefully some day we’ll have more on other geometries, too.)

   Does that mean they have more examples to add here? I saw your call for more examples had no response.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 13th 2011
   • (edited Jan 13th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexExamples for $\infty$-cohesive sites are still not abundant. But what I meant was that there are many other sites that large parts of the mathematical community care about (notably all variants of [[fppf site]], [[etale site]], etc.) whose $n$Lab treatment currently in no way reflects their relevance in the mathematical community, cohesive or not. Meanwhile, I have come to think that the route to more examples for cohesive $\infty$-toposes is slicing: for $\mathbf{H}$ a cohesive $\infty$-topos and $X \in \mathbf{H}$ geometrically contractible ($\Pi(X) \simeq *$) and [[small-projectve]], the slice $\mathbf{H}/X$ is at least locally and globally $\infty$-connected and local. (I am not yet sure about strong $\infty$-connectedness, i.e. $\Pi_X$ preserving products). I am thinking that to do cohesive [[derived geometry]] one ought to find a small-projective geometrically contractible $\infty$-stack $X$ "of $\infty$-functions". For the smooth case something like a stack $X : U \mapsto \{simplicial \; C^\infty rings over C^\infty(U)\}$. Then an object in $\mathbf{H}/X$ would be a smooth $\infty$-groupoid equipped with a structure sheaf of $\infty$-functions. And if $X$ could be tuned to be geometrically contractible and small-projective (not sure yet) then this slice topos would seem to be a decent candidate for derived smooth cohesive structure.

   Examples for $\infty$-cohesive sites are still not abundant. But what I meant was that there are many other sites that large parts of the mathematical community care about (notably all variants of fppf site, etale site, etc.) whose $n$Lab treatment currently in no way reflects their relevance in the mathematical community, cohesive or not.

   Meanwhile, I have come to think that the route to more examples for cohesive $\infty$-toposes is slicing: for $\mathbf{H}$ a cohesive $\infty$-topos and $X \in \mathbf{H}$ geometrically contractible ($\Pi(X) \simeq *$) and small-projectve, the slice $\mathbf{H}/X$ is at least locally and globally $\infty$-connected and local. (I am not yet sure about strong $\infty$-connectedness, i.e. $\Pi_X$ preserving products).

   I am thinking that to do cohesive derived geometry one ought to find a small-projective geometrically contractible $\infty$-stack $X$ “of $\infty$-functions”. For the smooth case something like a stack $X : U \mapsto \{simplicial \; C^\infty rings over C^\infty(U)\}$. Then an object in $\mathbf{H}/X$ would be a smooth $\infty$-groupoid equipped with a structure sheaf of $\infty$-functions. And if $X$ could be tuned to be geometrically contractible and small-projective (not sure yet) then this slice topos would seem to be a decent candidate for derived smooth cohesive structure.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJan 13th 2011
   • PermaLink
   Author: zskoda
   Format: MarkdownItexDoes this freedom in choice of "cohesive structure" give you a hint about the fiber interconnectedness which did not quite work in general in the Lie integration of higher Lie groupoids, for which you once created a mini-preprint "refined Lie integration" ?

   Does this freedom in choice of “cohesive structure” give you a hint about the fiber interconnectedness which did not quite work in general in the Lie integration of higher Lie groupoids, for which you once created a mini-preprint “refined Lie integration” ?

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeJan 13th 2011
   • (edited Jan 13th 2011)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItex>Also the term "discrete $\infty$-groupoid" would fit in nicely. This can be ambiguous, since it conflicts with [[discrete groupoid]] and (inasmuch as $\infty$-groupoids are spaces) [[discrete space]]. (In fact, these are the same kind of discreteness.) For Lie groupoids in particular, I've already learnt to distinguish ‘categorially discrete’ (the discreteness in my paragraph above) from ‘topologically discrete’ (the discreteness that you're referring to). So you should be able to use ‘discrete $\infty$-groupoid’, clarifying as ‘topologically discrete $\infty$-groupoid’ when necessary. (I see that this is already discussed at [[discrete category]].) On the other hand, inasmuch as $\infty$-groupoids are spaces, even ‘topologically discrete’ could be ambiguous in that case. All to say that while ‘discrete $\infty$-groupoid’ is certainly the correct term for this, one has to be careful.

   Also the term “discrete $\infty$-groupoid” would fit in nicely.

   This can be ambiguous, since it conflicts with discrete groupoid and (inasmuch as $\infty$-groupoids are spaces) discrete space. (In fact, these are the same kind of discreteness.)

   For Lie groupoids in particular, I’ve already learnt to distinguish ‘categorially discrete’ (the discreteness in my paragraph above) from ‘topologically discrete’ (the discreteness that you’re referring to). So you should be able to use ‘discrete $\infty$-groupoid’, clarifying as ‘topologically discrete $\infty$-groupoid’ when necessary. (I see that this is already discussed at discrete category.) On the other hand, inasmuch as $\infty$-groupoids are spaces, even ‘topologically discrete’ could be ambiguous in that case.

   All to say that while ‘discrete $\infty$-groupoid’ is certainly the correct term for this, one has to be careful.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJan 13th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, that's another clash of terminology. There are inevitably such clashes. But I have come to think that "categorically discrete groupoid" is best avoided in favor of 0-[[truncated]] groupoid. That is much more useful terminology, I think. So in the terminology that I prefer we have the following: * smooth $\infty$-groupoid that is a) 0-truncated and b) concrete $\simeq$ a diffeological space; * smooth $\infty$-groupoid that is a) 0-truncated and b) discrete $\simeq$ a set * smooth $\infty$-groupoid that is a) 1-truncated and b) concrete $\simeq$ a diffeological groupoid * smooth $\infty$-groupoid that is a) 1-truncated and b) discrete $\simeq$ a groupoid and so on

   Yes, that’s another clash of terminology. There are inevitably such clashes.

   But I have come to think that “categorically discrete groupoid” is best avoided in favor of 0-truncated groupoid. That is much more useful terminology, I think.

   So in the terminology that I prefer we have the following:

   • smooth $\infty$-groupoid that is a) 0-truncated and b) concrete $\simeq$ a diffeological space;

   • smooth $\infty$-groupoid that is a) 0-truncated and b) discrete $\simeq$ a set

   • smooth $\infty$-groupoid that is a) 1-truncated and b) concrete $\simeq$ a diffeological groupoid

   • smooth $\infty$-groupoid that is a) 1-truncated and b) discrete $\simeq$ a groupoid

   and so on

8. • CommentRowNumber8.
   • CommentAuthorTobyBartels
   • CommentTimeJan 13th 2011
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexGood point, although a lot of $0$-level terms have their own historical synonyms, and ‘discrete’ is just the one for ‘$0$-truncated’. So I use both, but no objection to taking the topological version as default in this context.

   Good point, although a lot of $0$-level terms have their own historical synonyms, and ‘discrete’ is just the one for ‘$0$-truncated’. So I use both, but no objection to taking the topological version as default in this context.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJan 13th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Does this freedom in choice of "cohesive structure" give you a hint about the fiber interconnectedness which did not quite work in general in the Lie integration of higher Lie groupoids In a way. I think this is handled by passing to cohesive over-toposes. In a cohesive $\infty$-topos there is abstract Lie theory for $\infty$-Lie algebras. To get the Lie theory for the $\infty$-Lie algebroids I think one needs to slice suitably over their base. To make this more concret, consider the cohesive $\infty$-topos for smooth $\infty$-groupoids. For $G$ a Lie group, we find $$ \mathbf{\flat}_{dR} \mathbf{B}G = \Omega^1_{flat}(-, \mathfrak{g}) \,, $$ where on the right we have the sheaf of sets of flat $\mathfrak{g}$-valued forms. The general abstract theory says that the [[Lie integration]] of $\mathfrak{g}$ is $$ \exp Lie \mathbf{B}G = \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \mathbf{B}G $$ hence $$ \exp Lie \mathbf{B}G = \mathbf{\Pi}_{dR} \Omega^1_{flat}(-) \,. $$ Now $\mathbf{\Pi}_{dR} X$ is defined to be the homotopy pushout $$ \array{ X &\to& * \\ \downarrow && \downarrow \\ \mathbf{\Pi} X &\to& \mathbf{\Pi}_{dR} X } \,. $$ Using the injective model structure, we may present this by the ordinary cokernel of the cofibration $$ \Omega^1_{flat}(-, \mathfrak{g}) \to \Omega^1_{fllat}((-) \times \Delta^\bullet, \mathfrak{g}) \,, $$ where on the right we have the path $\infty$-groupoid of the sheaf of forms in its concrete presentation by singular simplices. That's the crucial point here: on the right we have the simplicial presheaf that in degree $k$ has flat 1-forms on $(-) \times \Delta^k$. That's the $\infty$-groupoid whose objects are flat 1-forms, 1-morphisms are paths of infinitesimal gauge transformations, 2-morphisms are paths of such paths, and so on. The pushout identifies all the objects, i.e. forgets which 1-forms the gauge transformations act on. The result is indeed $\exp(\mathfrak{g})$ as described at [[Lie integration]]. The point of this is that 1. first we have forms on $(-) \times \Delta^k$ with no restriction; 1. then we kill of those forms with "legs" just along the $(-)$-direction. For $\infty$-Lie algebras this gives the expected result. For $\infty$-Lie algebroids its more subtle.

   Does this freedom in choice of “cohesive structure” give you a hint about the fiber interconnectedness which did not quite work in general in the Lie integration of higher Lie groupoids

   In a way. I think this is handled by passing to cohesive over-toposes.

   In a cohesive $\infty$-topos there is abstract Lie theory for $\infty$-Lie algebras. To get the Lie theory for the $\infty$-Lie algebroids I think one needs to slice suitably over their base.

   To make this more concret, consider the cohesive $\infty$-topos for smooth $\infty$-groupoids. For $G$ a Lie group, we find

   $$\mathbf{\flat}_{dR} \mathbf{B}G = \Omega^1_{flat}(-, \mathfrak{g}) \,,$$

   where on the right we have the sheaf of sets of flat $\mathfrak{g}$-valued forms.

   The general abstract theory says that the Lie integration of $\mathfrak{g}$ is

   $$\exp Lie \mathbf{B}G = \mathbf{\Pi}_{dR} \mathbf{\flat}_{dR} \mathbf{B}G$$

   hence

   $$\exp Lie \mathbf{B}G = \mathbf{\Pi}_{dR} \Omega^1_{flat}(-) \,.$$

   Now $\mathbf{\Pi}_{dR} X$ is defined to be the homotopy pushout

   $$\array{ X &\to& * \\ \downarrow && \downarrow \\ \mathbf{\Pi} X &\to& \mathbf{\Pi}_{dR} X } \,.$$

   Using the injective model structure, we may present this by the ordinary cokernel of the cofibration

   $$\Omega^1_{flat}(-, \mathfrak{g}) \to \Omega^1_{fllat}((-) \times \Delta^\bullet, \mathfrak{g}) \,,$$

   where on the right we have the path $\infty$-groupoid of the sheaf of forms in its concrete presentation by singular simplices. That’s the crucial point here: on the right we have the simplicial presheaf that in degree $k$ has flat 1-forms on $(-) \times \Delta^k$.

   That’s the $\infty$-groupoid whose objects are flat 1-forms, 1-morphisms are paths of infinitesimal gauge transformations, 2-morphisms are paths of such paths, and so on.

   The pushout identifies all the objects, i.e. forgets which 1-forms the gauge transformations act on. The result is indeed $\exp(\mathfrak{g})$ as described at Lie integration.

   The point of this is that

   1. first we have forms on $(-) \times \Delta^k$ with no restriction;

   2. then we kill of those forms with “legs” just along the $(-)$-direction.

   For $\infty$-Lie algebras this gives the expected result. For $\infty$-Lie algebroids its more subtle.

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 13th 2011
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    Author: DavidRoberts
    Format: MarkdownItexHow about 'bare $\infty$-groupoid for the topologically discrete version? This is what it truly is, because it is an $\infty$-groupoid with no other structure/cohesiveness.

    How about ’bare $\infty$-groupoid for the topologically discrete version? This is what it truly is, because it is an $\infty$-groupoid with no other structure/cohesiveness.

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJan 13th 2011
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    Author: Mike Shulman
    Format: MarkdownItexI'm mostly okay with "smooth ∞-groupoid" and "synthetic-differential ∞-groupoid," but not with "continuous" for something that's restricted to being modeled on cartesian spaces. Something like "topological-manifold ∞-groupoid" (although that is cumbersome) would be more accurate. I would expect "continuous" or "topological" to refer to something modeled on a site of *all* topological spaces, not just cartesian spaces.

    I’m mostly okay with “smooth ∞-groupoid” and “synthetic-differential ∞-groupoid,” but not with “continuous” for something that’s restricted to being modeled on cartesian spaces. Something like “topological-manifold ∞-groupoid” (although that is cumbersome) would be more accurate. I would expect “continuous” or “topological” to refer to something modeled on a site of all topological spaces, not just cartesian spaces.

12. • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 13th 2011
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItex@Mike 11 hear hear.

    @Mike 11 hear hear.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2011
    • (edited Jan 14th 2011)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> How about 'bare ∞-groupoid for the topologically discrete version? That's a possibility. I say "bare $\infty$-groupoid" a lot in expository text. i have mentioned this as an alternative at [[discrete infinity-groupoid]] now > not with "continuous" for something that's restricted to being modeled on cartesian spaces. Something like "topological-manifold ∞-groupoid" (although that is cumbersome) would be more accurate. Okay, I see what you mean. Maybe "manifold" is a bit awkward here but the term should refer to the specific topology of Euclidean spaces. What's that usually called? _Euclidean topology_ I suppose? So how about _Euclidean $\infty$-groupoid_ ? Or _Euclidean continuous $\infty$-groupoid_ ? Or _Euclidean topological $\infty$-groupoid_ ? An $\infty$-groupoid with _Euclidean continuous cohesive structure_ ?

    How about ’bare ∞-groupoid for the topologically discrete version?

    That’s a possibility. I say "bare $\infty$-groupoid" a lot in expository text. i have mentioned this as an alternative at discrete infinity-groupoid now

    not with "continuous" for something that’s restricted to being modeled on cartesian spaces. Something like "topological-manifold ∞-groupoid" (although that is cumbersome) would be more accurate.

    Okay, I see what you mean. Maybe "manifold" is a bit awkward here but the term should refer to the specific topology of Euclidean spaces. What’s that usually called? Euclidean topology I suppose?

    So how about Euclidean $\infty$-groupoid ? Or Euclidean continuous $\infty$-groupoid ? Or Euclidean topological $\infty$-groupoid ? An $\infty$-groupoid with Euclidean continuous cohesive structure ?

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2011
    • PermaLink
    Author: Urs
    Format: MarkdownItexre #6: I have edited and expanded [[discrete groupoid]] a bit more

    re #6: I have edited and expanded discrete groupoid a bit more

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeJan 14th 2011
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI'm okay with any of those "Euclidean" variants.

    I’m okay with any of those “Euclidean” variants.

16. • CommentRowNumber16.
    • CommentAuthorTobyBartels
    • CommentTimeJan 15th 2011
    • PermaLink
    Author: TobyBartels
    Format: MarkdownItexThe term "Euclidean" does not suggest the category $Top$ (as opposed to $Diff$, etc) to me. If anything, I would guess the category of [[Euclidean spaces]] and their manifolds, that is the one whose morphisms are distance-decreasing affine maps, on the grounds that historically these (or rather the invertible such) are the ones that Euclid recognised as preserving all structure (congruences in the sense of elementary geometry). On the other hand, "Euclidean continuous" and "Euclidean topological" are clear; now I know what the morphisms are. However, one could with equal justification say "Cartesian continuous"; [[Cartesian spaces]] come with even *more* structure than Euclidean spaces, but it doesn't matter how much structure we start with when the second term cuts most of it away.

    The term “Euclidean” does not suggest the category $Top$ (as opposed to $Diff$, etc) to me. If anything, I would guess the category of Euclidean spaces and their manifolds, that is the one whose morphisms are distance-decreasing affine maps, on the grounds that historically these (or rather the invertible such) are the ones that Euclid recognised as preserving all structure (congruences in the sense of elementary geometry).

    On the other hand, “Euclidean continuous” and “Euclidean topological” are clear; now I know what the morphisms are. However, one could with equal justification say “Cartesian continuous”; Cartesian spaces come with even more structure than Euclidean spaces, but it doesn’t matter how much structure we start with when the second term cuts most of it away.